Linear Quantile Regression Process Object

These are objects of class rq.process. They represent the fit of a linear conditional quantile function model.


These arrays are computed by parametric linear programming methods using using the exterior point (simplex-type) methods of the Koenker--d'Orey algorithm based on Barrodale and Roberts median regression algorithm.


This class of objects is returned from the rq function to represent a fitted linear quantile regression model.


The "rq.process" class of objects has methods for the following generic functions: effects, formula , labels , model.frame , model.matrix , plot , predict , print , print.summary , summary


The following components must be included in a legitimate rq.process object.


The primal solution array. This is a (p+3) by J matrix whose first row contains the 'breakpoints' \(tau_1, tau_2, \dots, tau_J\), of the quantile function, i.e. the values in [0,1] at which the solution changes, row two contains the corresponding quantiles evaluated at the mean design point, i.e. the inner product of xbar and \(b(tau_i)\), the third row contains the value of the objective function evaluated at the corresponding \(tau_j\), and the last p rows of the matrix give \(b(tau_i)\). The solution \(b(tau_i)\) prevails from \(tau_i\) to \(tau_i+1\). Portnoy (1991) shows that \(J=O_p(n \log n)\).


The dual solution array. This is a n by J matrix containing the dual solution corresponding to sol, the ij-th entry is 1 if \(y_i > x_i b(tau_j)\), is 0 if \(y_i < x_i b(tau_j)\), and is between 0 and 1 otherwise, i.e. if the residual is zero. See Gutenbrunner and Jureckova(1991) for a detailed discussion of the statistical interpretation of dsol. The use of dsol in inference is described in Gutenbrunner, Jureckova, Koenker, and Portnoy (1994).


[1] Koenker, R. W. and Bassett, G. W. (1978). Regression quantiles, Econometrica, 46, 33--50.

[2] Koenker, R. W. and d'Orey (1987, 1994). Computing Regression Quantiles. Applied Statistics, 36, 383--393, and 43, 410--414.

[3] Gutenbrunner, C. Jureckova, J. (1991). Regression quantile and regression rank score process in the linear model and derived statistics, Annals of Statistics, 20, 305--330.

[4] Gutenbrunner, C., Jureckova, J., Koenker, R. and Portnoy, S. (1994) Tests of linear hypotheses based on regression rank scores. Journal of Nonparametric Statistics, (2), 307--331.

[5] Portnoy, S. (1991). Asymptotic behavior of the number of regression quantile breakpoints, SIAM Journal of Scientific and Statistical Computing, 12, 867--883.

See Also


  • rq.process.object
Documentation reproduced from package quantreg, version 5.54, License: GPL (>= 2)

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